A Common Tangent Triangle

The parabola $y^2=2px$ ($p>0$) meets the hyperbola $y=-\dfrac 1x$ at a point $R$. A common tangent line of the two curves touches the parabola at $S$ and the hyperbola at $T$. Prove that for every positive real $p$, the area of $\triangle RST$ is a constant independent of $p$.